Pancharatnam Phase and Berry Phase

Abstract

The Indian physicist S. Pancharatnam, working in quantum Optics in 1956, introduced the novel concept of a geometrical phase. Let \(H(\xi )\) denote a Hamiltonian that depends on some parameters \(\xi \), with ground state \(|\psi (\xi )\rangle \). One can define a phase difference \(\varDelta \varphi _{12}\) between two ground states \(|\psi (\xi _{1})\rangle \) and \(|\psi (\xi _{2})\rangle \):

Parametric Hamiltonians convey the indirect influence of the rest of the Universe on the system under study, and this influence is somewhat similar to a special magnetic field in abstract parameter space.